Due: Wednesday, 29 July at 10:00 (submit at the start of class)
- Consider the following matrix:
a) Find three possible basis sets for the column space of U. The vectors in each basis should consist entirely of 0’s or 1’s.
b) For the system Ux = y are there any constraints on y such that a solution for x exists? Indicate yes or no and provide a brief explanation.c) Find a basis for the null space of U.
- Suppose the columns of a 5 x 5 matrix A are a basis for R5.
a) The equation Ax = 0 has only the solution x = 0 because ______________ [provide a brief explanation].
b) If b is in R5 then Ax = b is solvable because _______________ [provide a brief explanation].
- Consider the following matrix:
a) Compute the eigenvalues of A and a basis for the eigenspace of each eigenvalue.b) Compute A-1 and its eigenvalues.
c) Form and show the matrices necessary to diagonalize A.
d) Compute A4 using the diagonalization from part (c).
- Consider the following matrix:
a) Compute the eigenvalues for A. The characteristic polynomial should look simple, but solving it properly requires some understanding of the complex roots of unity.
b) Select one of the complex eigenvalues and determine a basis for its eigenspace (this is a pain, but its good for reviewing your complex number arithmetic).
Note that the answers for questions 3 and 4 can easily be verified using Matlab (although you are required to determine your answers by hand).